Graph extraction labelling and visualization

ABSTRACT

A technique is provided for modeling a network relationship via providing a graphical representation of the network relationship and labeling the graphical representation based on a symbol sequence. Further, one or more tools is provided by the present technique for interactively visualizing and/or analyzing the graphical representation of the network relationship based on the symbol sequence.

BACKGROUND

The invention relates generally to graphical modeling of a network andin particular to graphical modeling of a network based on symbolsequences.

A wide variety of techniques are employed to visualize and/or analyzevarious network relationship, such as blood vessel networks, neuralnetworks, electrical circuit networks, power system networks,communication networks transport networks or any other network. One ofthe popular techniques of representing a network relationship is byproviding a graphical representation of the network. Though thegraphical representation of the network relationship enablesvisualization of the network relationship and associated problems, theanalysis is typically performed manually.

For example, in a medical imaging application, a blood vessel network ismanually analyzed to identify potentially life-threatening aneurysms, orto plan surgical operations, or for other purposes. Alternatively,computer implemented techniques may be employed for the analysis of thegraphical representation of the network. However, these techniquesgenerally employ a brute force graph search algorithm that may becomputationally very intensive even for small graphs. In medical imagingapplications, atlases are useful frameworks for the representation ofanatomy derived from imaging data such as CT scan or MRI scan. Suchatlases are used for finding and comparing “normal” and “abnormal”regions. However, a limitation of such a method requires spatialcorrespondence between the atlas and the case being compared with. Thus,a spatial registration is performed between the atlas and the case undercomparison that is complex and prone to errors. An equivalent to anatlas in non-medical applications would include a standardized circuitdiagram. The current techniques are therefore cumbersome andinefficient.

It is therefore desirable to provide an efficient method forintelligent, automatic, and interactive visualization and analysis ofvarious network relationships. It is also desirable to provide agraphical representation of a network relationship that is spatiallyindependent.

BRIEF DESCRIPTION

Briefly, in accordance with one embodiment of the technique, a method isprovided for modeling a network relationship. The method includesproviding a graphical representation of the network relationship andlabeling the graphical representation based on a symbol sequence. Incertain embodiments, the technique may include systems and computerprograms that afford such functionality.

In accordance with another embodiment of the technique, a methodincludes extracting a planar graph from a n-dimensional data setrepresenting a network relationship. The method also includes performing(n-1)-dimensional connectivity analysis on each of two successive(n-1)-dimensional data sets to generate one or more (n-1)-dimensionallabels. The two successive (n-1)-dimensional data sets are extractedfrom the n-dimensional data set. The method also includes performingn-dimensional connectivity analysis on a portion of the n-dimensionaldata set comprising the two successive (n-1)-dimensional data sets togenerate one or more n-dimensional labels and determining one or moreedges and/or vertices of the network relationship based on the one ormore n-dimensional labels and corresponding (n-1)-dimensional labels. Incertain embodiments, the technique may include systems and computerprograms that afford such functionality.

In accordance with a further embodiment of the present technique, amethod is provided for labeling a planar graph. The method includesinitializing each vertex of the planar graph with a unique symbol from asymbol sequence. The planar graph includes a plurality of vertices andone or more edges between a pair of vertices. The method also includesassigning each edge of the planar graph with another unique symbolderived via an invertible composite function applied on all symbolsrepresenting a plurality of vertices from one or more source vertices ofthe planar graph to a destination vertex of the respective edge. Incertain embodiments, the technique may include systems and computerprograms that afford such functionality.

In accordance with an additional embodiment of the present technique, amethod is provided for modeling a network relationship. The methodincludes providing one or more tools for processing the networkrelationship based on a symbol sequence and providing a graphicalinterface for interactively visualizing and/or analyzing the networkrelationship via the one or more tools. In certain embodiments, thetechnique may include systems and computer programs that afford suchfunctionality.

In accordance with another embodiment of the present technique, animaging system is provided. The imaging system includes an imagingmodality for imaging a network relationship and providing an-dimensional data set representing the network relationship and acomputer coupled to the imaging modality. The computer is configured toextract a graphical representation of the network relationship from then-dimensional data set, label the graphical representation based on asymbol sequence and interactively visualize and/or analyze the graphicalrepresentation based on the symbol sequence.

DRAWINGS

These and other features, aspects, and advantages of the presentinvention will become better understood when the following detaileddescription is read with reference to the accompanying drawings in whichlike characters represent like parts throughout the drawings, wherein:

FIG. 1 is a flowchart illustrating an exemplary method of modeling anetwork relationship in accordance with one aspect of the presenttechnique;

FIG. 2 is a flowchart illustrating an exemplary method of extracting aplanar graph in accordance with one aspect of the present technique;

FIG. 3 is an example of graph extraction in accordance with the methodillustrated in FIG. 2;

FIG. 4 is a flowchart illustrating an exemplary method of labeling aplanar graph in accordance with one aspect of the present technique;

FIG. 5 is a flowchart illustrating the exemplary method of FIG. 4,including an example of an invertible composite function, in accordancewith one aspect of the present technique;

FIG. 6 is a graphical example of a generic labeled graph in accordancewith the method illustrated in FIG. 4;

FIG. 7 is a graphical example of a generic labeled graph in accordancewith the method illustrated in FIG. 5;

FIG. 8 is a graphical example of handling multiple edges between twovertices using a pseudo vertex in accordance with one aspect of thepresent technique;

FIG. 9 illustrates a blood vessel network as an example that can bemodeled in accordance with one aspect of the present technique;

FIG. 10 illustrates a computer network as another example that can bemodeled in accordance with one aspect of the present technique;

FIG. 11 illustrates a power distribution network as yet another examplethat can be modeled in accordance with one aspect of the presenttechnique;

FIG. 12 illustrates a electrical circuit network as a further examplethat can be modeled in accordance with one aspect of the presenttechnique;

FIG. 13 is a flowchart illustrating an exemplary method of pruning alabeled graph in accordance with one aspect of the present technique;

FIG. 14 is an example of a pruned network in accordance with methodillustrated in FIG. 11;

FIG. 15 illustrates an opacity transfer function for altering opacity ofdifferent portions of the network in accordance with one aspect of thepresent technique;

FIG. 16 illustrates a color transfer function for setting differentcolors to different portions of the network in accordance with oneaspect of the present technique;

FIG. 17 illustrates a system for modeling a network relationship inaccordance with one aspect of the present technique;

FIG. 18 illustrates an imaging system for visualizing and/or analyzing aphysiological network in accordance with one aspect of the presenttechnique;

FIG. 19 illustrates an example of identifying, analyzing and trackingthe progress of treatment in case of malignant tumor development; and

FIG. 20 illustrates an example of detecting and recording the path fromthe root to a blocked artery in case of a pulmonary embolism.

DETAILED DESCRIPTION

Embodiments of the present technique are generally directed towardgraphical modeling of network relationships based on symbol sequences.These modeling techniques may be useful in a variety of contexts where anetwork relationship needs to be visualized and/or analyzed via agraphical representation. For example, network relationships includeblood vessel networks, biological neural networks, electrical circuitnetworks, power distribution networks, communication networks, computernetworks, transportation networks (e.g., trains, airplanes, automobiles,and boats), flow networks (e.g., sewage, fresh water, oil, gas, ortraffic), or any other networks. Although the following discussion islimited to a few examples, a variety of networks and applications arewithin the scope of the present technique.

Referring now to FIG. 1, this figure is a flow chart that illustrates anexemplary process 10 of modeling a network relationship in accordancewith one aspect of the present technique. As illustrated, the process 10provides a graphical representation of a network relationship at block12. For example, the graphical representation of the networkrelationship may be predefined or it may be extracted from ann-dimensional data set representing the network relationship asdiscussed in further detail below. Further, embodiments of the graphicalrepresentation include an operational direction of the networkrelationship and one or more vertices and/or edges of the networkrelationship, as discussed in detail below. The graphical representationof the network relationship is further labeled based on a symbolsequence at block 14. For example, each of the vertices, edges, and/orsegments of the network relationship are labeled in a unique mannerbased on the symbol sequence. Examples of symbol sequence include anumber sequence, a string sequence, an alphanumeric sequence or a signsequence. In certain embodiments, the vertices, edges and/or networksegments are labeled with symbols that accumulate from one portion toanother within the network relationship. For example, each vertex may belabeled with a unique symbol, and then each edge or segment may belabeled with an accumulation of these unique symbols between andincluding the respective vertex and another vertex (e.g., a source)within the network relationship. The accumulation of these uniquesymbols may be comma delimited, additive, multiplicative, or may beachieved via concatenation or other mathematical and/or logicaloperations. Examples of string sequence include alphabets, words, orsentences. The number sequence may be either a super-additive numbersequence or a super-multiplicative number sequence. Examples of thenumber sequence include a prime number sequence, a Fibonacci numbersequence, a number sequence based on powers of a base number (e.g.,powers of base 2), or other super-additive or super-multiplicativenumber sequences. Examples of sign sequence include geometric symbols,mathematical symbols, scientific symbols or any other symbols. Further,one or more tools are provided at block 16 for visualizing and/oranalyzing the graphical representation of the network relationship basedon various operations on the symbol sequence. For example, as discussedin further detail below, these tools may include an opacity variationtool, a color variation tool, a pruning tool, or other graphicalprocessing tools that alter the graphical representation according tothe unique symbolical labeling of the network relationship.

The graphical representation of the network relationship may beextracted from an n-dimensional data set representing the networkrelationship via a variety of techniques. For example, FIG. 2 is a flowchart that illustrates an exemplary process 18 for automatic extractionof the graphical representation from an n-dimensional data set. In oneembodiment, the n-dimensional data set may be a binary data setrepresenting a blood vessel network or a neurological network. Theseexemplary data sets may be obtained from a wide variety of medicalimaging modalities, such as a Computed Tomography (CT) system, aMagnetic Resonance Imaging (MRI) system, an X-ray imaging system, andothers. As discussed in detail below, extraction of the graphicalrepresentation from the n-dimensional data set includes numericallyidentifying one or more vertices and/or edges of the networkrelationship along with a direction for each edge. It should be notedthat the vertices may be one of the five fundamental vertices, namelymerge type (M-type) vertex, branch type (B-type) vertex, source type(S-type) vertex, sink type (Z-type) vertex or null type (N-type) vertex.An edge can be defined as a segment that joins a vertex V_(i) withanother vertex V_(j) where i is not equal to j.

A merge vertex can be defined as a vertex where two or more incidentedges have a direction feeding into the vertex and at least one incidentedge has a direction feeding away from the vertex. A branch vertex canbe defined as a vertex where two or more incident edges have a directionfeeding away from the vertex and at least one incident edge has adirection feeding into the vertex. A source vertex can be defined as avertex where all edges that are incident have a direction away from thevertex and no incident edges have a direction feeding into the vertex. Asink vertex can be defined as a vertex where all edges that are incidenthave a direction into the vertex and no incident edges have a directionfeeding away from the vertex. A null vertex can be defined as a freevertex that has no edge incident upon it. A loop is combination of abranch type vertex and a merge type vertex on the same edge.

As illustrated FIG. 2, the exemplary process 18 begins at block 19 byproviding an n-dimensional data set representing a desired networkrelationship. At block 20, the process 18 further includes reading twosuccessive (n-1)-dimensional data sets from the n-dimensional data set.For example, the process 18 may include reading two successive rows orone-dimensional data set from a two-dimensional data set, whichcorresponds to a two-dimensional network image. The two(n-1)-dimensional data sets may be referred to as set R1 and set R2,where set R1 is superior to set R2. For example, the sets R1 and R2 maycorrespond to rows R1 and R2 or one-dimensional data sets from atwo-dimensional data set. At block 22, the process 18 continues byperforming an (n-1)-dimensional connectivity analysis on each of the twosets R1 and R2 independently to generate a set of labels L(n-1). Forexample, the process 18 may involve separately performing aone-dimensional connectivity analysis on each of the rows R1 and R2. Atblock 24, the process 18 further includes performing an n-dimensionalconnectivity analysis on a portion of the n-dimensional data setcomprising the two successive (n-1)-dimensional data sets {R1, R2} togenerate a set of labels L(n). For example, the process 18 may involveperforming a two-dimensional connectivity analysis on rows R1 and R2,taken together. For each n-dimensional label L(n) in set R2, adiscrepancy exists if any two data elements in set R2 having different(n-1)-dimensional labels L(n-1) is located at block 26. In case of sucha discrepancy, labels in R1 are labeled as a branch vertex at block 28.If there is no such discrepancy, labels in R2 are labeled as an edge atblock 30. Similarly, for each n-dimensional label L(n) in set R1, adiscrepancy exists if any two data elements in set R1 having different(n-1)-dimensional labels L(n-1) is located at block 32. In case of sucha discrepancy, labels in R2 are labeled as a merge vertex at block 34.If there is no such discrepancy, labels in R1 are labeled as an edge atblock 36.

Further, the process 18 checks for a first instance of R1 at block 38and labels all distinct L(n-1) labels in the first instance of R1 as asource vertex at block 40. If it is not the first instance of R1 and ifno labels L(n-1) are located in R1 at block 42, then all distinct labelsL(n-1) in R2 are labeled as a source vertex at block 44. Similarly, theprocess 18 queries for a last instance of R2 at block 46 and labels alldistinct L(n-1) labels in the last instance of R2 as a sink vertex atblock 48. If it is not the last instance of R2 and if no labels L(n-1)are located in R2 at block 50, then all distinct labels L(n-1) in R1 arelabeled as a sink vertex at block 52. Finally, the process 18 isrepeated for all of the portions of n-dimensional data sets indexed as aseries of (n-1)-dimensional data sets. The type of vertices may alsoprovide information regarding the operational direction of the networkrelationship. For example, in a blood vessel network or a powerdistribution network, the operational direction of the networkrelationship will be from a source vertex to a sink vertex that is fromheart to various organs in case of the blood vessel network or fromgenerating station to consumer units in case of the power distributionnetwork. However, it should be noted that in case of a computer networkthe operational direction depends upon flow of information betweennetwork devices.

FIG. 3 illustrates an example of graph extraction in accordance with theprocess 18 described above. In the upper left portion of FIG. 3, atwo-dimensional binary data set 54 is depicted. As described above, anytwo successive rows are picked at a time and a two-dimensionalconnectivity analysis is performed on a portion of the two-dimensionaldata set comprising the two successive rows to generate a set oftwo-dimensional labels L(2). In addition, one-dimensional connectivityanalysis is performed on each of the two successive rows independentlyto generate a set of one-dimensional labels L(1). These two sets oflabels L(2) and L(1) are compared with each other to extract graphicalinformation, such as the presence of various type of vertices and/oredges. For example, as illustrated in the lower right portion of FIG. 3,one-dimensional and two-dimensional connectivity analysis is performedfor each pair of successive rows such as row 1 and row 2, illustrated inthe lower left portion of FIG. 3, to generate correspondingone-dimensional and two-dimensional labels L(1) and L(2), respectively.A continuity in presence of data elements represented by ‘1’ is seenwhile performing two-dimensional connectivity analysis on a portion oftwo-dimensional data set comprising two successive rows, row 1 and row2. Hence, a set of two dimensional label L(2)={‘1’, ‘2’} is generated.On performing a one-dimensional connectivity analysis on row 1, thereexists a first and a second occurrence of continuous data elements.Further, on performing a one-dimensional connectivity analysis on row 2,there exists a third and a fourth occurrence of continuous dataelements. A set of one-dimensional labels L(1)={1,2,3,4} is generatedand assigned to the four independent sets of continuous data elements.The same labeling procedure is performed for the remaining rows in thetwo-dimensional binary data set 54.

Further, regarding rows 1 and 2, since row 1 is the first instance, alldata elements with distinct labels L(1) of row 1 are labeled as a sourcevertex. In addition, for each two-dimensional label L(2) in row 2, anytwo data elements in row 2 have the same one-dimensional labels L(1).Thus all data elements in row 2 having distinct labels L(1) are labeledas an edge. When comparing labels L(2) and L(1) for row 3 and row 2, adiscrepancy exists in row 2, because for each two-dimensional labelsL(2) in row 2, there are two data elements having differentone-dimensional labels in row 2. Hence all data elements in row 3 havingdistinct labels L(1) are labeled as a merge vertex. All data elements inrow 4 having distinct labels L(1) are labeled as an edge because nodiscrepancies in one-dimensional labels L(1) are located for eachtwo-dimensional label L(2) in row 4. However, on comparing labels L(2)and L(1) for row 5 and row 4, a discrepancy exists in row 5 because foreach two-dimensional label L(2) in row 5, there are two pixels havingdifferent one-dimensional labels in row 5. All data elements in row 4having distinct labels L(1) are therefore relabeled as a branch vertex.Further, on comparing labels L(2) and L(1) for row 6 and row 5, for eachtwo-dimensional label L(2) in row 5, any two pixels in row 5 has thesame one-dimensional labels L(1). Thus, all data elements in row 5having distinct labels L(1) are labeled as an edge. Finally, since row 6is the last instance all data elements in row 6 having distinct labelsare labeled as sink vertex.

Once the set of vertices and their type have been identified along withthe edges connecting two vertices, a graph 56 can be constructed asillustrated in the upper right portion of FIG. 2. As described in theabove example, the process 18 is repeated for all the portion oftwo-dimensional data sets indexed as a series of one-dimensional datasets to extract the graph 56.

The graphical representation of the network relationship thus obtainedis labeled based on a symbol sequence in accordance with an exemplaryprocess 58 depicted in FIG. 4. As illustrated in block 59, the exemplaryprocess 58 includes initializing each vertex of the planar graph with aunique symbol from a symbol sequence. At block 60, the process 58continues by computing another unique symbol derived via an invertiblecomposite function applied on all symbols representing all the verticesfrom one or more source vertices of the planar graph to a destinationvertex of an edge. The particular edge is then labeled with the computedsymbol. An invertible composite function is defined as a one-to-onemapping from the domain set of symbols to a range set of symbols suchthat a unique inverse exists. Thus, it is possible to recover thesymbols that were originally mapped to. Examples of invertible compositefunction include a multiplication function on a super multiplicativenumber sequence, an addition function on a super additive numbersequence, a union operation on the set of symbol sequence, or anothermathematical and/or logical operations.

As described in further detail below, whenever there are two or moreedges between the same pair of vertices, a pseudo vertex (P-type) isintroduced for each second and/or successive edge between the pair ofvertices in order to resolve conflict between the parallel edges. Eachpseudo vertex is assigned a unique symbol from the symbol sequence. Thesecond and/or successive edge is then labeled as described above usingthe pseudo vertex at blocks 61 and 62.

In one embodiment, the exemplary process 58 is further elaborated with a‘union operation’ as the invertible composite function. Referring now toFIG. 5, a process 63 includes initializing each vertex of the planargraph with a unique singleton set comprising a unique symbol from asymbol sequence at block 64. At block 65, the process 63 continues byidentifying a start and a stop vertex for each edge connecting twovertices with a particular direction of flow from the start vertex tothe stop vertex. Further, the process 63 includes computing a unique setfrom the union of one or more unique sets, which represent one or moreedges terminating at the start vertex of a particular edge and, also,the singleton set representing the stop vertex of that particular edgeat block 66. The particular edge is then labeled with the computedunique set. Alternatively, the unique set for a particular edge may beobtained by including all symbols representing the plurality of verticesfrom one or more source vertices of the planar graph to a destinationvertex of the respective edge. Further, another invertible compositefunction, such as multiplication in a prime number sequence, or additionin a super additive sequence, may be used to generate another uniquesymbol from all symbols of the computed unique set, which may then beused to label the edge. For example, when a prime number sequence isused for labeling the planar graph and the invertible composite functionused is multiplication, then LCM (least common multiple) of the labelsof all edges terminating at the start vertex is computed and the resultis multiplied with the stop vertex label to generate the label of theedge between the start and the stop vertex. The fact that every numbercan be uniquely represented as the multiplication of distinct primefactors makes ‘multiplication’ function a valid invertible compositefunction for a prime number sequence.

Further, each pseudo vertex is assigned another unique singleton setcomprising the unique symbol from the symbol sequence. The second and/orsuccessive edge is then labeled as described above using the pseudovertex at blocks 67 and 68.

As would be appreciated by one skilled in the art, the process 58 forlabeling the planar graph described above may be further extended toassign a unique code to each planar graph based on the unique set ofsymbols obtained above. Further, it should be noted that any two planargraphs may be compared with each other based on their unique code.

FIG. 6 illustrates an example of graph labeling in accordance with theprocess 58 described above. In the illustrated example, prime numbersequence is used for labeling the graph 70. Each vertex is labeled witha unique prime number. It should be noted that for the sake ofconvenience, ‘1’ is included in the list of primes={1, 2, 3, 5, 7, 11,13, 17, 19, 23, 29, 31, 37, 41, 43 . . . }. For each edge, the arrowindicates the direction of flow. The LCM of the labels of all edgesterminating at the start vertex of a particular edge is computed and theresult is multiplied with the label of the stop vertex of thatparticular edge, thereby generating a unique number. The particular edgeis then labeled with the generated unique number. As illustrated, thelabel of the edge terminating at vertex ‘2’ is “1.2”. The edge betweenvertex ‘2’ and ‘7’ is therefore “1.2.7” or simply “2.7”. Further, thetechnique become clearer when looked at vertex ‘31’. The labels of edgesterminating at ‘31’ are “2.7.11.23.31” and “2.7.13.29.31”. The edgebetween ‘31’ and ‘37’ is therefore labeled by a unique number that isequal to LCM of “2.7.11.23.31” and “2.7.13.29.31” times 37 or“2.7.11.13.23.29.31.37”.

FIG. 7 illustrates another example of graph labeling in accordance withthe process 63 described above. In the illustrated example, analphabetic sequence is used for labeling the graph 71. Each vertex islabeled with a unique alphabetical character. For each edge, a start anda stop vertex is identified. Further, a unique set is computed from theunion of labels of one or more edges terminating at the start vertex andthe singleton set representing the stop vertex for a particular edge.The particular edge is then labeled with the resulting unique set. Asillustrated, the label of edges terminating at {F} are {A, B, C, F} and{A, B, D, F}. The edge between {F} and {G} is therefore labeled by aunique set that is equal to union of {A, B, C, F}, {A, B, D, F} and {G}or simply {A, B, C, D, F, G}. Alternatively, in the illustrated exampleconcatenation operation may be used as an invertible composite function.The labels of the edges terminating at ‘F’ will then be ‘ABCF’ and‘ABDF’. The label of the edge between ‘F’ and ‘G’ is then obtained byconcatenating labels and ‘ABCF’, ‘ABDF’ and ‘G’ or ‘ABCDFG’.

An example of handling multiple edges between two vertices using apseudo vertex is illustrated in FIG. 8. As illustrated, a graph 72 hastwo parallel edges between vertex B and vertex M. A pseudo vertex P isintroduced in the second vertex between the vertex B and vertex M asshown in graph 74. The pseudo vertex P is assigned a unique number fromthe number sequence used for labeling the planar graph. The graph 74 isthen labeled in accordance with the technique 58 described above.

As would be appreciated by one skilled in the art, the techniquesdescribed in above embodiments for graph extraction and graph labelingare not dependent on each other. A predefined graph may be provided andlabeled in a similar manner while modeling a network relationship.Alternatively, the graph may be extracted and then labeled whilemodeling the network relationship. Further, the graph may simply beextracted in accordance with the technique and may be used forvisualization and/or manual analysis of a complex network relationshipand its associated problems. Further, it would be appreciated by oneskilled in the art that the techniques described above may be used in awide variety of network relationships.

For example, FIG. 9 illustrates a blood vessel network 76 with itsvertices and/or edges identified and labeled. The heart or other bloodproducing organs acts as a source while the different organs receivingblood may act as a sink. The blood vessels such as arteries, veins,capillaries are the edges with its complex network where they may mergeat a merge vertex or may branch out at a branch vertex. The operationaldirection is decided based on the blood flow in various organs.

FIG. 10 illustrates another example of a computer network 78 such as alocal area network (LAN), a wide area network (WAN) or at least a partof the Internet. As illustrated, the computer network 78 may include awide array of devices such as a desktop terminal, a printer, a router, alocal server, a central server, a LAN and/or other network devices, allconnected to the network. These network devices may be labeled intovarious types of vertices and the connections between them may belabeled as edges connecting various devices. The operational directionmay be decided based on the information flow between theseinterconnected network devices.

FIG. 11 illustrates yet another example of a power distribution network80 for transmitting electrical power from a generating station tovarious users. As illustrated, the power distribution network 80 mayinclude a generating station, a transmission station, a distributioncenter and/or consumer units such as industrial units, residentialapartments and/or colonies, commercial establishment, small scaleindustries, railways and others. The generating station may be labeledas a source vertex while the consumer unit may be labeled as a sinkvertex. The transmission station may be labeled as a branch or a mergevertex depending upon the number of distribution center it is connectedto or number of generating station it is connected to. Similarly, thedistribution center may be labeled as a branch or merge vertex. Thetransmission wires connecting these units may be considered as edges ofthe network relationship and the operational direction is decided basedon the power flow from the generating station to the consumer units.

FIG. 12 illustrates yet another example of an electrical circuit network82. As illustrated, the electrical circuit network 82 may include a widevariety of circuit components such as one or more power sources,resistors, capacitors, inductors, electrical loads and/or othercomponents. These circuit components may be labeled into various typesof vertices and the connections between them may be labeled as edgesconnecting various vertices. The operational direction may be decidedbased on the flow of electrical current between these circuitcomponents.

Once the planar graph representing the network relationship is labeled,the network relationship may be interactively visualized and/or analyzedvia one or more tools based on the operation on the symbol sequence.These one or more tools may include a pruning tool to limitvisualization of the network relationship, a compositeness tool toidentify a degree of connectivity of one portion to other portions ofthe network relationship, a roundness tool to identify a number ofconnections between two portions of the network relationship, an opacitytool to alter opacity of different portions of the network relationship,and/or a color visualization tool to set one or more colors to differentportions of the network relationship. A graphical interface may beprovided for interactively visualizing and/or analyzing the networkrelationship via the one or more tools mentioned above.

FIG. 13 illustrates an exemplary process 84 for pruning the graphicalrepresentation of a network relationship based on the symbol sequence.Based on a degree of pruning required, maximum and/or minimum allowableconnectivity (N) is identified for visualization and/or analysis for aparticular graph at block 86. For each edge, the number of elements (N1)in the unique set representing the respective edge is then computed atblock 88. Finally, the opacity of all the edges having N1>N (for maximumallowable connectivity) and/or N1<N (for minimum allowable connectivity)is reduced at block 90. In one embodiment, where a prime number sequenceis used for labeling the planar graph representing the networkrelationship, the number of divisors (N1) of all the edges iscalculated. The opacity of all the edges having N1>N (for maximumallowable connectivity) and/or N1<N (for minimum allowable connectivity)is then reduced to obtain a pruned graph. The unique labeling techniquedescribed above preserves the hierarchy of the pruned graph irrespectiveof their sizes or intensities. Thus an edge with a larger numeric labelthat is farther from the root and hence is less important than a vesselwith a smaller numeric label gets pruned.

FIG. 14 illustrates an example of pruning on a portion of the planargraph 92 in accordance with the process 84 described above. The planargraph 94 is a pruned version of the portion of the planar graph 92 withmaximum number of divisors 2, while the planar graph 96 is a prunedversion of the portion of the planar graph 92 with maximum number ofdivisors 3.

The number of elements in the unique set representing a particular edgedetermines the degree of compositeness of that particular edge. In casewhere prime numbers are used to label the planar graph, the number ofdivisors of the label of the particular edge determines the degree ofcompositeness of that particular edge. Edges having a high degree ofcompositeness indicate that they have more number of connections toother edges in the planar graph. This may indicate the dependency orredundancy of a particular edge. For example, an edge with label of 180has 17 divisors thereby indicating that the edge may be potentiallyconnected to 17 other edges with those particular labels. Suchinformation may be useful in many ways. For example, in case of bloodvessel network or neurovascular network it may be useful to a doctor toknow the dependency or redundancy of a particular blood vessel or neuronwhile planning for surgery. Similarly, in a power system network, it maybe useful for a technician to know the dependency or redundancy of aparticular transmission line during fault analysis or maintenanceoperations.

Roundness is defined as the sum of the powers of the prime factors inthe decomposition of a particular label. For example, the roundness ofan edge having the label 72=2³X3² is 3+2=5. This indicates the number ofconnection that exists between two edges. Higher roundness implies thatthere is more redundancy between two particular edges. In the aboveexample, there are three connections between edges having labels 72 and2, while there are two connections between edges having labels 72 and 3.

Further, the present technique includes different transfer functions tocontrol the visualization of the graphical representation of the networkrelationships based on the respective labels of the edges. For example,an opacity transfer function 98 as illustrated in FIG. 15 may be used toalter opacity of different portions of the network relationship based ontheir labels. As illustrated, the opacity transfer function 98 sets theopacity of edges represented by labels ‘70’, ‘110’, ‘1430’, and ‘1870’to zero. This is a useful tool if we are interested in visualizing andanalyzing a specific part or region of the network. As would beappreciated by one skilled in the art, the opacity transfer function 98may be used while pruning a planar graph. Referring back to the planargraph 94 of FIG. 14, the opacity of all the edges with the number ofdivisors greater than 2 may be reduced, while in the planar graph 96opacity of all the edges with the number of divisors greater than 3 maybe reduced.

In addition, a color transfer function 100 as illustrated in FIG. 16 maybe used to set different colors to different portion of the networkrelationship based on their labels. The tool can be useful invisualizing different portion of network in different colors under aparticular situation. As illustrated, the color transfer function 100sets the color of edges represented by labels ‘1’, ‘3’, ‘69’ and 2967 toblue, the color of edges represented by labels ‘14’ and ‘157251’ togreen and the color of edge represented by label ‘175053’ to red. Forexample, suppose there is a fault in one of the transmission lines in apower distribution network or a cut in one of the blood vessel in ablood vessel network. A technician or a doctor will be interested inknowing the branches that are affected by the particular fault or a cut.This fault or cut is represented as an edge having a particular label inthe planar graph representing a network relationship. For prime numbersequence based labeling with invertible composite function asmultiplication of prime factors, all edges that are affected by thefaulty edge will have labels that are a multiple of the label of thefaulty edge. Similarly, all edges feeding into the faulty edge will befactors of the label of the faulty edge. Any other edge having a labelthat is relatively prime to the label of the faulty edge will remainunaffected by the fault or the cut. This information can be easily shownvia different colors set via the color transfer function 100. Forexample, edges feeding into the faulty edge may be set with a blue colorand edges that are affected by the faulty edge or more precisely edgesthat are fed by the faulty edge may be set with a red color. All otheredges that are not directly affected by the faulty edge may be set witha green color.

It should be noted that depending on the complexity of the network andthe symbol sequence used, the values of the labels obtained may tend tobe quite large, thereby leading to overflow problems. This problembecomes is prominent for any super-multiplicative number sequences. Fora very large network, such as those of the neurovascular, the number ofvessel and interconnections may exceed 100, the prime factors getmultiplied and the values of the label bloat exponentially. Suchsituations may be avoided by taking logarithm to any base that mayresult in smaller numbers and convert the operation of multiplication tothat of addition. Alternatively, a super-additive number sequence suchas Fibonacci number sequence or any base-N number sequence may be usedthat employs addition operation or some other symbol sequence may beused.

All the above mentioned techniques may be employed through any generalpurpose computer for modeling of a network relationship. As illustratedin FIG. 17, a system 102 for modeling a network relationship includesmodules for graph extraction 104, graph labeling 106, and graphvisualization and/or analysis 108. These modules may be presentseparately as a special purpose circuitry 110 or may be present as apart of the computer. The system 102 may further include or be incommunication with memory 112. It should be understood that any type ofmemory 112 to store a large amount of data may be utilized by such anexemplary modeling system 102. As illustrated, the graph extractionmodule 104 may process the n-dimensional data set representing thenetwork relationship residing in the memory 112 and extract a graphrepresenting the network relationship in accordance with the techniquedescribed above. The extracted planar graph may then be stored in thememory 112 or may be further processed by the graph-labeling module 106.The graph-labeling module 106 may then label the extracted graph inaccordance with the technique described above. Alternatively, apre-loaded graph may be loaded from the memory 112 and labeled by thegraph-labeling module 106. The labeled graph may then be stored in thememory 112 or may be processed by graph visualization and/or analysismodule 108 for interactively visualizing and/or analyzing the graph. Aswould be appreciated by one skilled in the art, a pre-stored labeledgraph may be processed by the graph visualization and/or analysis module108 for the purpose of visualization and/or analysis.

In addition, the system 102 may be configured to receive commands andscanning parameters from an operator via an operator workstation 114.For example, the operator workstation 114 may be equipped with akeyboard and/or other input devices by which an operator may control themodeling system 102. Thus, the operator may visualize and/or analyze thenetwork relationship and its associated problem via an interactivegraphical interface, initiate modeling, and so forth.

A display 116 may be coupled to one of the operator workstation 114 andmay be utilized to visualize and/or analyze the network relationship andto control visualization via the interactive graphical interface.Additionally, the graphical representation of the network relationshipmay also be printed on to a printer 118 that may be coupled to theoperator workstation 114, either directly or over a network. It shouldbe further noted that the system 102 and/or operator workstation 114 maybe coupled to other output devices that may include standard or specialpurpose computer monitors and associated processing circuitry.Furthermore, additional operator workstations may be further linked inthe system 102 for outputting system parameters, performingvisualization and/or analysis and so forth, so that more than oneoperator may perform visualization and/or analysis.

In one embodiment, an imaging system 120 as illustrated in FIG. 18includes the system 102 for modeling the network relationship describedabove coupled to one or more diagnostic imaging system 122, such as a CTsystem, an MRI system, an X-ray imaging system, a PET system and others.These diagnostic imaging systems 122 collects n-dimensional data setrepresenting the blood vessel network or neurovascular network via dataacquisition circuitry 124. The n-dimensional data set is then stored inthe memory 112 of the modeling system. Alternatively, an imagereconstructor 126 may reconstruct the image from the n-dimensional datasets and the image is stored in the memory 112 of the modeling system.The blood vessel network or the neurovascular network is then modeledvia the modeling system 102 as described above.

The modeling of the blood vessel network and/or neurovascular networkvia the imaging system 120 described above may help in visualizationand/or analysis of many medical problems, such as for identifyingpotentially life-threatening aneurysms, or planning surgical operations,or for other purposes. For example, as illustrated in FIG. 19, a tumoris identified and monitored via the imaging system 120 described above.In the top portion of FIG. 19, this figure illustrates a top of thebrain 128 that acts as a sink vertex under a normal condition. Agraphical representation 130 of the blood vessels in this particularregion is extracted and labeled in accordance with the techniquesdescribed above. If a malignant tumor 132 develops in this region, thetumor 132 will act as sink and will draw large amounts of blood fromsurrounding regions. Accordingly, new arteries will be formed feedingblood to the tumor 132 in the region 134. The number of arteries leadinginto the tumor (sink vertex) 132 can be computed by finding the degreeof compositeness, thereby identifying and classifying the tumor 132 inthe region 134. Compositeness is an indicator of degree of arterialsupply. Malignant tumor 132 can therefore be differentiated through ahigher compositeness factor. The middle portion of FIG. 19 illustrates agraphical representation 136 of the blood vessel in this particularregion having malignant tumor 132 before the treatment. The method maybe used to track the effectiveness of therapy. If sometime later thearterial tree has changed and the compositeness is lower aftertreatment, as illustrated in the lower portion of FIG. 19 via thegraphical representation 138, it can be concluded that arterial supplyis reducing and hence the tumor 132 is dying.

FIG. 20 illustrates another example of visualizing and/or analyzing apulmonary embolism via the imaging system 120 described above. Asillustrated, pulmonary embolism is a condition that occurs when anartery 140 in lungs is blocked due to one or more blood clots 142 thattravel to lungs from another part of the body. In some cases, othertypes of clots such as globules of fat, air bubbles, tissue from a tumoror a clump of bacteria may also lead to blockage. Detection of the pathof the blocked artery 140 from root 144 is generally tedious and isbased on trial and error. However, as illustrated, once the blockedartery 140 has been identified, the imaging system 120 automaticallyfinds and displays the path 146 from the root 144 to the blocked artery140 based on the labeled graphical representation. This improves thediagnosis and treatment of the pulmonary embolism, while reducing therisk to the patient. Further, the topology and segmental relationshipmay be recorded for future reference.

The graph extraction, labeling, visualization and/or analyzingtechniques described in the various embodiments mentioned above provideintelligent, automatic and interactive visualization and analysis ofvarious network relationship. Intelligent labeling of the graph by usingsymbol sequence is such that connectivity information is embedded, inthe unique symbol representing the edges of the network relationship. Onthe fly automatic adaptation of visualization parameters such as colorand opacity for highlighting an edge and its parent/child braches. Asopposed to computationally intensive brute force graph search, thetechniques described in the various embodiments discussed above allowsinstantaneous access to parent and children of selected vessel throughprime decomposition of the label. Computation of the prime factors is ofvery low computational burden and can be further speeded up by using alook-up-table that lists all the labels and their corresponding primedecompositions. The look-up-table is very useful where the network issmall. On the whole, the above mentioned techniques results in improvedcomputational efficiency. In addition, the extracted graphicalrepresentation of the network relationship is spatially independent asit just carries information about the topology of the network.

As will be also appreciated, the above described techniques may take theform of computer or controller implemented processes and apparatuses forpracticing those processes. The disclosure can also be embodied in theform of computer program code containing instructions embodied intangible media, such as floppy diskettes, CD-ROMs, hard drives, or anyother computer-readable storage medium, wherein, when the computerprogram code is loaded into and executed by a computer or controller,the computer becomes an apparatus for practicing the invention. Thedisclosure may also be embodied in the form of computer program code orsignal, for example, whether stored in a storage medium, loaded intoand/or executed by a computer or controller, or transmitted over sometransmission medium, such as over electrical wiring or cabling, throughfiber optics, or via electromagnetic radiation, wherein, when thecomputer program code is loaded into and executed by a computer, thecomputer becomes an apparatus for practicing the invention. Whenimplemented on a general-purpose microprocessor, the computer programcode segments configure the microprocessor to create specific logiccircuits.

While only certain features of the invention have been illustrated anddescribed herein, many modifications and changes will occur to thoseskilled in the art. It is, therefore, to be understood that the appendedclaims are intended to cover all such modifications and changes as fallwithin the true spirit of the invention.

1. A method of modeling a network relationship, the method comprising:providing a graphical representation of the network relationship byextracting the graphical representation from an n-dimensional data setrepresenting the network relationship using a modeling system; andlabeling the graphical representation using the modeling system based ona symbol sequence, wherein the symbol sequence comprises asuper-additive number sequence or a super-multiplicative numbersequence.
 2. The method of claim 1, further comprising identifying anoperational direction of the network relationship.
 3. The method ofclaim 1, further comprising identifying one or more vertices and/oredges of the network relationship.
 4. The method of claim 3, whereinlabeling the graphical representation comprises referencing each vertexof the network relationship with a unique symbol from the symbolsequence.
 5. The method of claim 3, wherein labeling the graphicalrepresentation comprises referencing each edge of the networkrelationship with a unique symbol derived from an invertible compositefunction applied on all the symbols representing a plurality of verticesfrom one or more source vertices of the network relationship to adestination vertex of the respective edge.
 6. The method of claim 1,wherein labeling comprises sequentially uniquely referencing originationand termination points of segments of the network relationship with aunique symbol from one or more segments to one or more hierarchicallyarranged segments in the network relationship.
 7. The method of claim 1,wherein labeling comprises referencing a desired segment in a set ofsuccessively connected segments of the network relationship with a setof unique symbols, wherein the set of unique symbols includes a uniquesymbol for each of the origination and/or termination points for thedifferent segments in the set of successively connected segments.
 8. Themethod of claim 1, wherein the symbol sequence further comprises anumber sequence, or a string sequence, or an alphanumeric sequence, or asign sequence.
 9. The method of claim 1, wherein the super-additivenumber sequence or the super-multiplicative number sequence comprises aprime number sequence, or a Fibonacci number sequence, or a numbersequence based on powers of a base number.
 10. The method of claim 1,further comprising providing one or more tools for interactivelyvisualizing and/or analyzing the graphical representation of the networkrelationship based on the symbol sequence.
 11. The method of claim 10,wherein the one or more tools comprises a pruning tool to limitvisualization of the network relationship based on the symbol sequence.12. The method of claim 10, wherein the one or more tools comprises apruning tool to limit visualization of the network relationship based ona hierarchy based pruning of the labeled graphical representation. 13.The method of claim 10, wherein the one or more tools comprises acompositeness tool to identify a degree of connectivity of one portionto other portions of the network relationship based on the symbolsequence.
 14. The method of claim 10, wherein the one or more toolscomprises a roundness tool to identify a number of connections betweentwo portions of the network relationship based on the symbol sequence.15. The method of claim 10, wherein the one or more tools comprises aopacity tool to alter opacity of different portions of the networkrelationship based on the symbol sequence.
 16. The method of claim 10,wherein the one or more tools comprises a color visualization tool toset one or more colors to different portions of the network relationshipbased on the symbol sequence.
 17. The method of claim 1, wherein thenetwork relationship comprises a blood vessel network, or a biologicalneural network, or an electrical circuit network, or a powerdistribution network, or a wiring network, or a communication network,or a computer network, or at transportation network, or an air trafficnetwork, or a road network, or a network of rivers and/or tributaries,or a flow network.
 18. A computer readable storage media, comprising:code adapted to provide a graphical representation of a networkrelationship, wherein the code adapted to provide the graphicalrepresentation of the network relationship comprises code adapted toextract the graphical representation from an n-dimensional data setrepresenting the network relationship; and code adapted to label thegraphical representation based on a symbol sequence, wherein the symbolsequence comprises a super-additive number sequence or asuper-multiplicative number sequence.
 19. The computer readable storagemedia of claim 18, further comprising code adapted to provide aplurality of tools for interactively visualizing and/or analyzing thegraphical representation of the network relationship based on the symbolsequence.
 20. A system for modeling a network relationship, the systemcomprising: means for providing a graphical representation of thenetwork relationship by extracting the graphical representation from ann-dimensional data set representing the network relationship; and meansfor labeling the graphical representation based on a symbol sequence,wherein the symbol sequence comprises a super-additive number sequenceor a super-multiplicative number sequence.